Find n if ^ n - 1 P_3 :^n P_4 = 1:9 - Vidyadip

Question

Find n if $^{n - 1}{P_3}{:^n}{P_4} = 1:9$

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Answer

Here $^{n - 1}{P_3}{:^n}{P_4} = 1:9$
$\therefore \frac{{(n - 1)!}}{{(n - 4)!}}:\frac{{n!}}{{(n - 4)!}} = 1:9$.
$ \Rightarrow \frac{{(n - 1)!}}{{(n - 4)!}} \times \frac{{(n - 4)!}}{{n!}} = \frac{1}{9}$
$ \Rightarrow \frac{{(n - 1)!}}{{n!}} = \frac{1}{9} \Rightarrow \frac{{(n - 1)!}}{{n(n - 1)!}} = \frac{1}{9}$
$ \Rightarrow \frac{1}{n} = \frac{1}{9} \Rightarrow n = 9$

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